In prior attempts for correcting non-liner relation between an input and an output values in image output machines, Bezier curves, polynomial curves and dividing curves are each used as a gamma (.gamma.) correction curve. For example, a third Bezier curve as a .gamma.-correction curve is expressed in the following equation: EQU B=(1-t).sup.3 P.sub.0 +3(1-t).sup.2 tP.sub.1 +3(1-t)t.sup.2 P.sub.2 +t.sup.3 P.sub.3 (1)
where B is an output result, Pi's are control points, t is a parameter (0.ltoreq.t.ltoreq.1). In order to alter the output results based upon a Bezier curve, the control points in the equation (1) are controlled. However, since the Bezier curve does not pass through the control points, the above described modifications are not generally intuitive and require trial and error approaches.
To facilitate the above described difficulty, the Bezier curve has been modified (herein after a modified Bezier curve or MB curve). A third MB curve is generally defined as follows: EQU MB=(1-t).sup.3 P.sub.0 +(cP.sub.1 -(c-3)P.sub.0)(1-t).sup.2 t+(dP2-(d-3)P.sub.3)(1-t)t.sup.2 +t.sup.3 P.sub.3 (2)
where curvature parameters c and d generally define curvature and c is an incline at the beginning while d is an incline at the end. When c=3 and d=3 in the above equation (2), the MB curve converge on a corresponding Bezier curve.
Now referring to FIG. 1, various MB curves are illustrated by modifying the curvature parameters c and d without modifying the control points. According to the above equation (2), after the curvature parameters c and d are determined, the output curve is primarily determined. Thus, two of the beginning inclination c, the ending inclination d, the coordinate (c-d)/8 at x=0.5 are determined. In order to determined other coordinates, a coefficient is changed to a polynomial. For example, c=c1+c2(1-t) where d=.+-.c, three additional points other than the beginning and ending points are determined.
Theoretically speaking, by increasing the power of an polynomial, the number of passing points is also increased.
In applying a MB curve, assuming that P.sub.0 =(0, 0), P.sub.1 =(0, 1), P.sub.2 =(1, 0) and P.sub.3 =(1, 1), the output y is defined as follows: EQU y=c(1-t).sup.2 t+(3-d)(1-t)t.sup.2 +t.sup.3 (3)
t in the equation (3) is replaced by x in the following equation (4). EQU y=c(1-x).sup.2 x+(3-d)(1-x)x.sup.2 +x.sup.3 (4)
in other words, input x and output y are normalized and each range between 0 and 1. According to the equation (4), a curve becomes dy/dx.vertline.x=0=c and dy/dx.vertline.x=1=d, yx=0.5=0.5+(c-d)/8 where c is a slope at x=0 and d is a slope at x=1. (c-d)/8 expresses a distance from x=0.5. In other words, c and d respectively determine the slope at the beginning and the end while c-d determines a middle point.
Referring to FIG. 2, a number of the gamma-correction curves is illustrated by varying the curvature parameters c and d. The gamma correction curves have a common middle point, a common beginning point and a common ending point while other points are varied among the curves.
In the above described and other prior attempts, it is generally difficult to determine an outcome or results of the gamma correction in an intuitive manner. For approximated curves such as a collection of straight line segments require a large number of control points and tend to have errors in direction at joining points. On the other hand, rational polynomials generally require increased computation and complex expression. Lastly, Bezier curves require complex computation such as differential due to parameter while MB curves additionally require parameters c and d which are not intuitive in defining overall shape of the curves.